The Constitution's ineligibility clause prohibits Senators and Representatives from also serving in the Executive or Judicial Branch. Has there ever been a situation where a Senator or Representative served in state government concurrently with service federally?
I don't know what case law says, but here's the United States constitution, Article I, Section 6, paragraph 2:
No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been encreased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.
So this would depend on whether a state office is an office under the United States or not. There may be case law (precedent) on this point.
Each state may also have rules. The National Conference of State Legislatures lists a summary of the rules for each of the fifty states. I'm not going to try to include all of them, but here's Alabama:
Federal office holders may not hold a state office, and state officials may not hold two offices at once. Exception for justices of the peace, constables, notaries public, and the commissioner of deeds. Ala. Const. Art. XVII, § 280 & Ala. Code § 36-2-1.
Federal office holders are specifically excluded from state offices. So even if the federal constitution or law does not prevent it, the Alabama state laws do for Alabama office holders.
While serving in the legislative assembly, no member may hold any full-time appointive state office established by this constitution or designated by law. During the term for which elected, no member of the legislative assembly may be appointed to any full-time office that has been created by the legislative assembly. During the term for which elected, no member of the legislative assembly may be appointed to any full-time office for which the legislative assembly has increased the compensation in an amount greater than the general rate of increase provided to full-time state employees. N.D. Const. art. IV, § 6.
North Dakota's summary does not say anything about federal offices. Whether that's a failure of the summary or an accurate depiction of North Dakota law, I don't know.
These examples are not meant to be representative. Alabama is the first one I checked. North Dakota was the first that I found that did not start something like "No person" nor include federal office holders at least implicitly. I didn't attempt a careful reading though, so I could have missed more examples like North Dakota. I found several other examples like Alabama while I was looking for North Dakota.