Alright, let's see if I can tackle this question a bit better than just saying "don't do it."
First, allow me to rephrase your question to what I believe you actually mean:
What's the worst possible starting hand that can still be played profitably given optimum conditions?
To answer that, I'll revert back to my answer above: Suited connectors gain in value against many opponents, and non suited-connectors like shorthanded pots. I think it's easier, then, if we separate the two:
Suited connectors
Given optimum conditions - a full table where everyone limps in and I know there will be no raise - I will play 32s. I won't flop a straight or a flush very often, but when I do, the pot will make up for small bets I pay to get there. Especially if I get the nut straight; whoever is holding an ace will pay me off dearly. So when it comes to suited connectors, I could go all the way down.
Non-suited non-connectors
A non-suited, non-connector wants to play heads-up, for largest chance to win. So, to fabricate a heads-up situation where you haven't already gotten a discount in the form of a blind, let's say that you're on the button in a limit hold'em tournament, the small blind is in the bathroom, and you know that the big blind never raises preflop, and never folds preflop. These conditions are important, because we're looking for the optimum: If he could fold preflop, you should play weaker hands yet for a raise, because you have some steal
equity. If he could raise preflop, you should tighten up, because you don't want your marginal hands to have to pay two bets to see the flop.
What's the worst hand you should play here? The hand that's in the 51st percentile - the hand that figures to be
just better than his average hand. I'm not sure exactly where this line goes, but I'll guess it's somewhere around J8o.
Given that both of these scenarios are hypothetical and not actually applicable to just about any decision you'll ever actually make, I feel compelled to add this: Why do you ask?