On legacy reports in Site Explorer, the option of "Full export" can be grayed out on the Legacy Backlinks report and not clickable.

This is because the amount of rows is above your plan limit:

On new reports, you'll see that the number of export rows is limited by plan.

You can use filters to create a report with less rows, and export full list of results multiple times.

Higher plans will have higher rows per report limits, allowing you to export more results in a single file. Here are the rows export limitations for all 4 plans:

Beyond your plan's base export limits, additional export rows per month can also be purchased on a pay-as-you-go basis.