Legally, it is going to depend upon your country's embassy, the random embassy, the extradition agreements in force between the two nations, the current political conditions between the two nations at the time the request for extradition is both made and executed, and the crime for which you are being sought. Extradition laws are as complex as they are numerous. If there is no extradition treaty or agreement between the two nations, then there is no mechanism to return you to the nation that seeks to arrest you. For US extradition law, see 18 USC 3181.
Practically, it is going to depend upon your country's embassy, the random embassy, the extradition agreements in force between the two nations, and the current political conditions at the time you are in the embassy. It is not impossible that in times of upheaval one nation might invade and seize the embassy of another.
There have been many attacks on embassies and diplomatic missions, and certainly not all of them have been accidents. One of the most infamous was the seizure of the U.S. Embassy in Iran in 1979. The movie "Argo" made famous "The Canadian Caper" in which six American embassy workers escaped into the streets of Tehran, and hid out in the Canadian, British, New Zealand, and I believe Swedish embassies--though they spent the most time in the Canadian embassy. Had they been discovered, the embassy would have been likely attacked and everyone inside taken hostage. Subterfuge and exfiltration were necessary to get them out of Iran safely.
There have been a few high profile cases over the years in which a person sought refuge within an embassy. Manuel Noriega, József Mindszenty, and Morgan Tsvangirai are a few. Mindszenty was granted political asylum at the United States embassy in Budapest, and lived there for 15 years before being allowed to leave Hungary. Just because one is safely inside an embassy, even if there is little to no risk that the host nation would lay siege to the embassy, one cannot smuggle out a person inside of a diplomatic pouch. Once you leave the physical property of the embassy, one is again on the soil of the host nation and subject to their laws.