On the turn. You need around 20% equity to make the call, however factoring in implied on you can call wider. As you are calling off half of your stack your implied odds of hitting the straight wont change much as you dont have much left behind in terms of spr, however as you have not stated what the size of the pot was on the turn i cant say wether its profitable or not. Instead i will give you an example hand to help you understand.
Say we get to the turn with our OESD pot is $2000 we have $1600 behind and villian covers us. If villian bets $800 here we have to call $800 to win $3600 giving us an equity of 22% as our chances of hitting OESD are around 16% this is not profitable. Thus we fold although with our remaining stack we have implied odds to call because it will be pofitable if we hit. We have to look at our situation if we miss and we will be left in a toug spot with a small amount of chips more than 80% of the time