Welcome to the Forum, Tinque:
I don't know how much of the old stairs you need to replace but, if its just the treads, you can buy them at a local lumber yard; just cut them to length, glue and nail in place. Let us know how it turns out.
The treads need to be a hard Yellow pine, standard grade shelving 1xs are not strong enough. Oak is even better. Depending on how your staircase is built, you may need to access them from underneath to do the repair correctly. Not knowing how your staircase is assembled makes it hard to help you much further.