Can a dining table be repurposed if the legs won't hold it up anymore?

Sure, but you can also get new legs for the table if you wish to continue using it.

Perhaps the legs could be shored up so they will be able to support the table top. I would see if it repairable. If not, perhaps you could find legs to replace them. You could go a little shorter on the legs and turn it into a gaming table for playing board games or cards, or weather proof it and turn it into a patio table if it is not too big.

You can take the old legs off and replace them. This link tells you now. If you want to use it for something else, you can do that, too.

https://www.google.com/search?q=refinishing+an+old+iron+tub&rlz=1C1BLWB_enUS700US700&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiN582dndLWAhXq0FQKHS5tB5wQ_AUICygC&biw=1024&bih=666

I'd consider wrought iron legs, sanding and resurfacing the top if it's lost its glitz and keep on using it. Another option is do you have, or could you accommodate, a kitchen island? Really pretty wood would add pizzazz. Enough coats of poly would allow it to stand up to lots of moisture but not heat.

A gentleman in england cut his table in half and used it as walls in a camper remodel. Beautiful results

If your table is wood, we cut our old table in half, and secured it to a wall which I use for sewing, make new legs or support it with a 45 degree angled piece of 2x4.