Can I install 125 AMP rated sub-panel when the need is 60 AMPS
Please install a bigger box!!!

Your box definitely should exceed minimums in this area: Number of spaces.   Listen, there's always a future project, there's always another load.   The out-of-pocket cost for a larger panel is very small (especially when you consider other factors, as we're about to).  But the regrets and unnecessary expense when you run out of spaces is a rather expensive affair.  

As such we very strongly recommend you wildly oversize your panels.  You need 8 spaces? Consider nothing less than 24 spaces.  All of our panels here have plenty of extra space (except an 8-space Pushmatic that's a nightmare) and that's the way we like it.  I myself would go 30-space. 

I mean "space" not "circuit" - you see a lot of 20 space/40 circuit panels - that's only a 20-space.  

You wouldn't drive 85 mph on 85 mph-rated tires, would you?

Suppose you lived in Nevada where posted limits are 80 mph. Yeah, you'll want the 115 or 130 mph tires I think :) 

The same applies to service panel amp ratings.  You wouldn't run 60A on a 60A box, would you?  Of course not, you want some safety headroom.  So 125A or even 200A is perfectly reasonable.  It's a "limit" thing, not a "mandatory match" thing. 

This also ties into the "spaces" problem because I gather you were feeling stifled with only 8 spaces.  As well you should!  You are at liberty to get as large a panel as you want. 

You need a main breaker there!

To be more precise, you need a main disconnect switch.  But the cheapest most practical way to get that is to use a box with a main breaker.  

Now I happen to know none of the 8-space panels on the market include a main breaker. So setting up one of those "straight" would be a code violation.  If there are 6 or fewer throws, "no main breaker" can be legal, but never with 8.   

You might skate by the "Rule of Six" if two of your breakers are 2-pole. However that "paints you into a corner" of being unable to add a seventh throw in the future.  And you know you will! 

Voltage drop - yikes

Obviously, voltage drop will be a big problem.  At 2 AWG wire and 48A draw (figuring for 80% of your 60A breaker that you seem happy with), your voltage drop will be 7.04%, which is high even by my standards.  

That would not be acceptable in Canada or in a Tim Horton's restaurant.  In Canada you'd be required to use 4/0 AL wire, which is ridiculous in my opinion, eh?  

I haven't priced 600' of that stuff but I'm sure I will not like the price.  

By the way, if you aren't in Canada, you are allowed to breaker the feed breaker for whatever the wire limits are, e.g. 90A for #2 aluminum.  

Transformers!

Like I say, I haven't priced the fat wire, but it may be worth looking at transformers for the long haul.   The wire is good for 600V (and 600V transformers are readily available in Canada), but around the States 480V is more likely to be findable cheap used.  If you're buying new transformers then price 600V. 

In this case, you use 2 transformers back to back, 480V-jumpered primaries facing each other.  This steps up the transmission voltage to 480V.  Now, the wire gets a lot thinner (for the same voltage drop).  

If you do it Canada-style, holding voltage drop to a firm 3% at 80% of rated current, then #4 aluminum will suffice, eh? From 4/0 down to #4, that's the power of transformers for 'ya!  If you don't mind that 7% drop from your original proposal, are you sitting down?  #8 aluminum or #10 copper gets it home.  

So transformers let you drop 6-8 numerical sizes. More if they are 600V transformers. 

However, I would recommend #6 aluminum at that point, giving a very acceptable 4.29% drop at 48A practical draw (80% of breaker trip).

You probably won't push the wire to its thermal limits.  The reason is the transformers are a limiting factor also.  I'm imagining you get 15 KVA transformers which will limit you to 60A@240V total throughput and you'd have to breaker for that at the supply side to protect the transformer.  You might get a better deal on used 25 KVA transformers, which would let you got 100A @ 240V.  If that's the plan, a bump to #4 Al wire might be called for. 

Obviously transformers are not free, but they make the wire a lot thinner, so you have to crunch the numbers and see what you get.