Weren't they betting for you? You've got just under a 30% chance of making this at the turn & river (taken separately). Almost any flop bet is callable, obviously.

If you're going to bet, you either want it to be less than the odds of you catching say 1/2 pot on the flop and 1/3 on the turn, or you just want to push and rely on the "it'll come more than half the time" odds.

I probably wouldn't do either and would just call it, since (as you said) they aren't laying their hands down. This way if you do miss completely it only costs as much as the bets you called, instead of costing more. If you might have been able to bluff them out, that could be a different story though.