I have an old road bike I want to get a single speed crank for but don't know what fitting the bottom bracket has.
As ojs says, you have a spindle for cottered cranks. Cottered cranks basically do not exist anymore, unless you look for 'vintage' parts. To fit a new single speed crank you need to replace the bottom bracket bearings and axle along with the crank.
Your bottom bracket shell is am internally threaded kind. The most commonly encountered standard for thread width and pitch is known as 'English' or 'BSA'. On an older bike this may not be the case.
If your bike is English threaded there are lots of options available. Probably the best option is a threaded internal cartridge BB bearing with square taper or splined crank interface. Square taper single speed cranks are plentiful and inexpensive. This also allows you to choose the spindle length to get the chainline you need.
If you can get the old bearings out and ca get hold of an old English threaded BB you could try it for size. I say this a bit hesitantly though because I have a feeling there are some threadings that are close to but not exactly the English standard.
Update: Sheldon Browns page on threaded BB shells for reference: https://www.sheldonbrown.com/cribsheet-bottombrackets.html