YES, although it is highly unlikely.
The presidential candidate must be:
A. A natural-born citizen of the United States,
B. At least 35 years of age, and
C. A resident of the United States at least 14 years.
Source: U.S. Const., art. II, § 1(5)
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Do you think Obama would be president had he been busted for smoking pot and snorting lines ? Now thats a question you should be asking . But to answer your question , Yes , you can still run for president with a prior felony .
Given that he has most likey committed a felony in order to get impeached and convicted, it is likely most states would deny him the right to vote. If he is convicted or pleads guilty to a crime prior to impeachment, he would likely lose his voting rights in most states for a period of time. See Sources and related links for additional information.
It does not say that in the Constitution. Felony firearm rights are a matter of Federal Law and state laws that say a person convicted of a felony shall not possess, use, transport, sell, purchase, carry, ship, receive, or distribute firearms.
Be at least 18 years old - Apex
Basically, it means you can't, and yes, you have lost some rights, but not necessarily for life. It is possible to have your rights restored, but for information on that, you really should contact an attorney familiar with the process.