If it is a DVD modded or optical drive emulator (ODDE) then yeah it should be fine as long as it is not otherwise banned. DVD mods did used to ban people but that has not happened in years, and that mainly leaves people doing silly things with profile modding or getting banned because they are idiots online or something.
If it is a JTAG or RGH modded 360 (these are the ones with region free, USB, XBLA, DLC, modded games, cheats....) then consider it a no. There is some limited scope if you decide to have two boxes and share profiles or have a dual nand and fiddle but it is not going to be a simple matter.
There are no real other mods still active, and they would not allow achievements to be assigned to your profile anyway.