Can I put a modern mattress on it?

Yes, you can fit a new mattress on an antique frame. You often can't use off-the-shelf sizes. The old imperial measurements rarely match standard modern equivalents. Many companies now specialise in making custom mattresses for historic frames. You just need to measure the internal width and length accurately before ordering.

I have an old iron bed but some of the wooden slats underneath are missing and I was wondering if any timber yard can cut new ones to fit?

Any timber merchant can cut new slats for you. Pine is the standard choice. Take one of the surviving slats with you so they can match the thickness and length. Make sure the new boards sit flat on the side rails. If they are too thick, the mattress will sit unevenly. You want them spaced a few inches apart to allow air to reach the mattress.

Will an antique bed take my weight?

A well-made antique single bed is structurally solid. The original timber or iron joints were built to last. Check the corner brackets and side rails. As long as the frame sits square and the bolts are tight, it will support a standard adult safely.

My grandmother gave me this wooden bed from her old house but the side rails are quite short so I am not sure an adult can actually sleep comfortably in it.

Shorter frames are common in Victorian and Edwardian furniture. People were generally shorter, and beds were built to fit the smaller rooms in many nineteenth-century houses. You can sometimes extend a wooden bed by having a cabinetmaker splice new timber into the side rails. This involves extending the internal ledges as well. Most dealers suggest leaving the frame original and using it in a child's room instead.