If a line like this __would obviosly have a gradient of 0, what would a line of | be?

A vertical line has "no slope". You could say it has "infinite slope" since if you have a line y = mx + b, the line approaches a vertical line as you let m go to ±∞. In any case, your change in y can be anything for the same change in x, so the idea of slope or gradient isn't necessarily applicable.