Starbound – Can You Mine Through the Entire World?

On the Starbound forums there are several complaints about the lack of this very feature. See a couple here and here (note the forums seem under a lot of stress at this time so you may have to reload a couple times).

It appears that as of this time in Beta Stage 1 - Progenitor on 12/7/13 there is no pause feature, but I was not able to find an official developer response to this.

UPDATE (5/11/14): It appears that this has still not been implemented in the game, but I looked for a work-around when a comment was made. This link led me to a suggestion that you always keep a chair around. Apparently you do not get hungry while you are sitting (or apparently sleeping) so you can wall yourself in and just take a seat. I put this to the test and it seems to work.

I took this picture before going to lunch and ate up to full at 12:29PM (Sorry if that's not readable). I made lunch for myself and ate and came back to see if my little Stitch was still doing alright. I clicked the Alt key to see how he was doing at 1:05pm and:

Still full. I left him a fire so as not to freeze to death, but with this chair and a few blocks I feel this could suffice for a pause-like feature. You might also consider carrying a bed with you so you can not only not starve, but also heal.

I've seen vines growing deep underground, well beyond the reach of 'root' plants. And I haven't ever noticed them growing back after being chopped down.

So to answer your question, no, sadly you cannot do anything to speed up the harvest of plant fibers from roots. There is no plant that you can move to increase the speed of root-growing. They simply exist, and stop existing once chopped down.

You could go to another planet and see if it has leafy trees though. Large plants (tree-sized) that have a less woody-looking construction drop a lot of plant fiber, which is great for making lots of bandages.