This question comes up a lot, and is then followed by replies that say "that doesn't make any sense". And, strictly speaking, it doesn't make sense. Scenes are just switcher patches that control which blocks are on and which are off (plus channel selection) for a particular scene. (Confusion over how scenes work is also a frequent subject on the forum.)
Usually what people mean when they ask this is: can I save a set of blocks into a "sub-preset" that I can drop it into another preset later, and in that way copy a group of blocks from one preset to another?
That's again a question that comes up a lot, and the answer is again "no". You can only copy or save a single block. But, this is a more important issue with the Axe-FX than on other modelers. With the Axe-FX, you can build some interesting meta effects by grouping together blocks that interact. For example, you can build complex delays that have filters or other effects of your choosing in the feedback loop. Or have a set of parallel effects that blend together. As it is, you must craft those from scratch each time you want to use them in a preset.
It would be nice if you could select a group of blocks, by drawing a lasso around them in the editor, then copy them, or even save them into a library. This is yet another case where the Axe-FX has a really cool feature that could benefit from even more powerful features.