Sometimes in Site Explorer, the option of "Full export" is grayed out on the Legacy Backlinks report and not clickable because the amount of rows is above your plan limit:
On Backlinks 2.0, you'll see that the number of export rows is limited by plan.
Luckily, you can download a huge part of them by specifying the exact number.
Here are the rows export limitations for all 4 plans:
Related: