Get yourself an old Remington 510. Single shot, automatic safety, pinpoint accurate. A more perfect rifle was never made.
Agree with Jackal. I've seen a lot of old Remington "Teeners" (510s & 514s, in particular) on tables at the last few gunshows.
Decently priced in the $100-to-$150 range. Not sure why they're popping up now, but they're out there, along with a few old single-shot Winchesters.
All the ones I saw were irons only.
I have my Dad's 510 Target Master that he got for his 12th birthday (now scoped), and a 514 that was fitted w/ a peep sight.
Both are super accurate, although the 514 has the better trigger.
From a ".22 family" picture: