Felons are allowed to join the military. It makes it very difficult but it’s not impossible. It greatly depends on the type of offense and your conduct since the offense. Sexual Offenses, domestic violence convictions that make it impossible to carry a weapon, and drug distribution will disqualify you. Something like, pulling this out of a hat, falsifying records to hide your affair with a pornstar would not make it impossible to join. You would have to get a waiver in order to join. How successful you are at getting a waiver depends on several factors. One of the most important factors is how badly they need bodies at that particular moment.
Equally unqualified for that, as well.