Which disadvantages does Russia fear?
There are multiple reasons, both tactical and strategic.
The main ones are:
Basically, Russia has no natural defensive perimeter of its core. Thus, its permanent strategy is to surround itself with satellite states which provide defense in depth and natural defensive perimeters of their own. This is elaborated extensively on in Statfor writeups on Russia, if you want a more detailed look.
If Ukraine is integrated into the West, Russia sees it as a strategic military threat - there's very little defense should Ukraine be a base of attack as opposed to a defensive buffer.
In addition, Ukraine is being made an example of (as was Georgia earlier, in a somewhat different game). Other neighbors can see that they have more to lose by cutting ties with Russia than by making ties with the West, since Russia has both political resolve and effective methods to punish or reward.
Economics of gas and oil.
Ukraine is where a big pipeline from Russia to its energy clients in Europe goes through. Russia needs to control that pipeline.
In addition, there are Gazprom's rights to develop and sell Ukrainian energy resources.
UPDATE May 2014: This item was posted in Dec 2013... now, 6 months later, we have the news that the annexed Crymean oil and gas resources are being nationalized and given for free to GazProm. (src). My Nostradamus badge is pending.
Russian goods would have to compete with EU goods. Without EU deal, if Ukraine is a member of "Customs Union", Russian goods have an advantage in Ukrainian market
Ukraine is basically a country split into two distinct demographics - Russia oriented ones (there is a large minority of 17.3% of ethnic Russians in Ukraine) and Europe oriented ones, especially in Western Ukraine. Russian national policy as announced by Putin is to protect "Russians" everywhere, Ukraine included.
Desire to screw things up for the West
Overall, Russian psyche - and its government reflects that - is firmly in "world politics is a zero sum game" area. If something is good for the West/US, it must be bad for Russia, even if there aren't obvious downsides, so "sticking sticks into the wheels" (to use the Russian expression) of any Western-beneficial plans is something they can't pass up. See also: their support for Al-Assad (who is, objectively speaking, of VERY little tactical or strategic value to Russia, the port lease deal could be done with any of his successors); or for Saddam.
 E.g. last surveys showed nearly 50/50 split about EuroMaidan pro-EU-integration movement; and ~50/50 on EU vs RF integration. ref
I would summarize it the following way.
Basically, Prometheism. Prometheism is an anti-Russian ideology invented in Poland during the 1910s and whose aim was the disintegration of the Russian Empire and, later, the Soviet Union. The Poles aimed to help any separatist movements inside Russia and to support anti-Russian attitude among Russian neighbours. Prometheism is still ongoing. For example, on November 22, 2007, in Tbilisi, Georgia, a statue of Prometheus dedicated to Polish President Lech Kaczyński was opened by anti-Russian Georgian President Mikheil Saakashvili. In the 1990s Russia experienced a bloody war with separatists in Chechnya and the separatists at the time were supported by anti-Russian forces from Ukraine.
Fear of a "Color Revolution". A set of the West-inspired color revolutions happened in Ukraine, Kyrgyztan, Georgia, attempted in Belarus and Russia. More bloody rebellions happened in Libya, Egypt and Syria. The rebellious winners in Libya after taking power provided help to a similar insurrection in Syria. Russia does not want this to happen in Ukraine again and does not want it to become a step for exporting the revolution to Russia.
Fear of possible revision of the consequences of WWII. Currenty the opposition forces who rally against the government in Ukraine, admire the pro-Nazi Ukrainian fighters of WWII. This may lead to eventual re-habilitation of Nazism, as it happened in the Baltic states. In the Baltic states, Russian language speakers (up to 40% of the population) were stripped of citizenship (in violation of Article 15 of the Universal Declaration of Human Rights) and apartheid regimes were instituted. Re-evaluation of the outcome of WWII and rehabilitation of Nazism can lead to territorial claims to Russian territory and demands for material reparations.
Economic connections between Russia and Ukraine. There are multiple and dense connections between Ukraine and Russia. For example, most of the planes designed by the Ukrainian design bureau Antonov are produced in Russia, while those produced in Ukraine use Russian parts. On the other hand, many of the Russian helicopters, including the military ones, use Ukrainian-made engines. Ukraine is also an important transit country for Russian oil and gas to Europe. The practice showed that after anti-Russian regimes were installed in the Baltics, the transit through those countries became next to impossible.
Fear of NATO expansion. Expansion of NATO will make Russia vulnerable to an attack by NATO forces. In recent years, NATO showed it is not self-restrained from attacking independent countries, such as Yugoslavia and Libya. Also, the Russian military industry would be unable to use the Ukrainian parts legally as produced by a probable adversary.
Competing projects of Russia-centered economic blocks on the post-Soviet space. Participation in both the Customs Union and the EU at the same time is impossible. For example, German Kanzlerin Angela Merkel has recently underlined that Ukraine cannot participate in two customs unions at the same time. Absence of Ukraine in the Customs Union may undermine the very idea and make it less attractive to other CIS countries.
Personal connections and benefits for Russian and Ukrainian citizens. There are multiple mixed families between Russians and Ukrainians and many perceive the both people as parts of one due to common culture and history. The association of Ukraine and the EU may complicate crossing the Russian-Ukrainian border and break other ties. For example, as a Russian citizen, I received from the Russian government free recreation tours to Ukraine, to Crimea and Odessa before the first Color Revolution. After it happened, such tours became unavailable due to stopped cooperation between the governments. Still Crimea remains one of the most popular vacation places for Russians.
Everything is about money and power of course. Nothing about "democracy", "joining to EU", and other media bullshit.
1) Most of Ukrainian economic is a part of Russian economic. At least we bought most of Ukrainian industry's production. With war and without Russian market Ukraines GDP falls ~50% down last 2 years. Just imagine - falls down ~two times!
2) Russia had mutual open markets with Ukraine. If Ukraine would open its markets to EU this would lead to uncontrolled traffic of EU productions on Russian market through Ukraine (without any standard taxes). This really means discriminating Russian production on its own market. EU and Ukraine wins, Russia loses. Why you think we want that?
3) Ukraine want both open markets with Russia and with EU. They want to make some margin on EU-Russia trade. EU wants that too. The actual problem is they want to make business on Russian loses, without respecting Russian interests.
4) Ukraine is under partial control of US politics. At least Ukraine government can make decisions, valuable for US but hurting Ukraine itself. The most important was the solution to stop rent of Crimea naval bases by Russia in 2017 and provide these bases to US. That decision would make Russian naval forces value in Black sea down to zero.
I don't understand why US, EU, and Ukrainian politics decided we will allow all this crazyness to become reality. Of course not. This is combined attack on Russian interests. And we will defend our markets and our naval fleet in Black sea (which is actually Mediterranean naval fleet) by all means.
There are balance of power considerations. For instance, modern Russia has only 140 million people. If the old Polish-Lithuania Commonwealth (from the Middle Ages) were reconstituted with Poland, the Ukraine, Belarus, and the Baltic states, it would consist of a nation of about 100 million people that would be a threat to Russia.
To further illustrate the point, in 1941, there were about 85 million Germans (counting Austrians, Ssudeten German in Czechia, Danzigers, etc.) and 170 million Soviets, of which about half were Russian, and half were non-Russian. If Germany had cast its invasion of the Soviet Union as a war of "liberation" (and claimed, we are only out to liberate the non Russian Soviet states from Russia), it could have created a buffer zone of Ukraine and other non Russian states that would form a strong alliance against "Russia."
Put another way, the Ukraine has historically been Russia's "safety belt" vis-a-vis Europe. Allied with America, Europe, Germany, or some other power against Russia, it could be very threatening.