Applicants from a wide range of different backgrounds and professions are hired into the federal government. I have worked alongside federal agents who were anthropologists, small business owners, software designers, school teachers, morticians, letter carriers, and local/state law enforcement. In other words, you don't have to have a law enforcement background to be picked by a federal agency. Every federal agency is different as far as what knowledge, skills, and abilities they look for in an applicant. Your best course is to contact the agencies you want to work for and talk to a recruiter. If I were in your shoes, I would take the job that was offered since you saw it as an"offer too good to turn down." As you will see, being processed by a federal law enforcement agency can take anwhere from 1 to 3 years. Hope this helps and good luck.
Eunice: A .22? Oh you've gotta be kidding me. That's like bringing a knife to a gunfight.
Detective Greenly: Yeah, or bringing a really small gun... to a gunfight.
Boondock Saint II