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:

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