The disavow tool isn't going to "fix" these 404s.
404's aren't always a bad thing. The warnings in GWT are just there to make you aware that there's potentially a problem with your site. It doesn't mean there IS a problem.
Is there content on your site that visitors clicking on these links should be arriving at? In which case you want to implement 301 redirects so that your visitors arrive on the most appropriate pate.
If there's nothing relevant on the site any more - a 404 error is perfectly acceptable.
Of course, you want to make sure that your 404 page gives the visitors the best chance/incentive to dig into the content on your site. Adding a nice obvious search box and/or links to most popular content may be a good idea. If you're getting lots of visitors from a particular site that you can maybe tailor your 404 message depending on the referrer.
The drawback here is that links pointing at 404 error pages won't pass link-equity. If there is value in the links, and you're happy that they're going to be seen a natural/authentic as far as google is concerned then you can always 301 redirect these.
Where you really should pay attention is where you have internal links on your site that are reporting 404s. These are under your control and you really don't want to give you visitors a poor experience with lots of broken links on your site.