Best way to store massive amounts of data

On my website, I currently have a giant form which now has about 700 fields. I have one column in my database for every field, and every row is a new form entry.

Over the last few years, this form has been filled out a few thousand times. This form is modified quite frequently, and the database has become very difficult to manage.

Just opening up this database in phpMyAdmin takes about a minute. I set up indexing so querying isn't slow, but this database is just not manageable.

I'm wondering what the best way to store large amounts of data, like this, would be? Is there a better practice, like using two tables and storing the users field entries in different rows?

How would you build this database structure? Would you recommend using NoSQL?