If I have a page hidden based on a condition, can the questions on that page be required?

If they are required, what happens if the conditions never take the user to that page?

The user will be asked to fill out that required field only if he reaches the page, if the never reaches the page due to conditions, then the field will not trigger the validation. 

You should reconsider to change the field's position or show it always in case you need to always have a value submitted there.