Lots of CRM training...

If there are any women among the crew, you have one more problem to consider; what happens if someone gets pregnant? Let's face it, humans are humans; over a ten-year span in such conditions, sex is basically a given. Even if you pack a ten-year supply of contraceptives (every pound of weight matters in spaceflight, so that's not trivial), you still rely on the assumption that people will use them properly. Having the women be post-menopause might help, but the accompanying physical changes tend to take a toll on fitness, which you probably can't afford on a long-term spaceflight.

Any answer to this would be pure speculation. There is currently not enough understanding of human psychology to answer this correctly. We've had a few experiments simulating flight to Mars, which is 6 times shorter than your specified time, and those experiments already provided conflicting results and went against expectations of professional psychologists.

There are a lot of old SF dealing with this issue. This does seem a time where it's more important to have stable people rather than trained astronauts. And ten years without "cuddles" is uncomforable, but not fatal.

Let there be an imposter among them

Send math grad students!

It would take quite a bit of work to get me to suspect my disbelief enough to believe it can take 60 times longer to do something despite still having more than half the workforce available. Unless perhaps everyone needs to, for example, work together to lift the same heavy object or perform some other heavy physical labour together (but even then 60 times seems like a stretch). Or if they can only work for 1 month every 5 years because the rest of the time the environment is not habitable or accessible.

Explorers using wooden sailing ships a couple centuries ago, and arctic explorers about a century ago did spend years on the sea with no (or next to no) contact with the rest of mankind, and they still managed fine. And they had even less entertainment options than what we could now provide.

@Palarran, I think vasectomies would solve that problem (and there is a decent chance that a reversal procedure would be successful once the astronauts are back on earth).

Reminds me of the book To be Taught, if Fortunate by Becky Chambers. I highly recommend you read this if you haven't already as it's rather relevant to your question.

@Palarran aside from permanent contraception, you could also send only post-menopausal women. In fact there are significant advantages to a crew of retired people (not the same thing but very compatible) and I might write an answer on that basis later

What they did in 2001 -- suspended animation.

An untrained civilian crew exploring exoplanets... they're expendable but they don't yet know it. The actual purpose of the mission is to retrieve a specimen... presumably by becoming the specimen... it's all buried in the 4000-page contract.

I remember a really good, old book that sent a group of scientists to a star for years of travel. They all ended up learning everything, changing partners, one died and became a ghost, and all become beyond super geniuses. I won't reveal anymore for spoilers, but I can't remember the title or author, or Google the plot. Great book. I want to say Fred Pohl or Asimov wrote it, but I just don't remember.

It all comes down to respect.

Daily mind-wipes a-la MIB flashy sticks. Assuming all ship log events are recorded the crew could give themselves a mind-wipe at bedtime and wake up with a whole new bunch of people to interact with. This should nip any romantic - and also abrasive - relationships in the bud.

Couldn't the sexual stuff be solved by just chemically castrating the crew?

@vsz The explorers typically spent at most a few years at sea, while these people will spend 2 decades. And while the explorers had a whole ship to wander around and dozens to hundreds of other sailors to interact with, these will be confined to a tiny spaceship (if the spaceship is even somewhat realistic, it will be tiny, as increasing size by even a little requires increased fuel for acceleration and deceleration, then you need more fuel to carry the fuel, and more fuel to carry the fuel to carry the fuel, and so on (tyranny of the rocket equation)).

Also, why would a crew stay behind, even if the world is a "Garden of Eden"? No matter how beautiful and stuff it is, you can't get any of the benefits of civilization there. So no new entertainment, no company, no support, no healthcare, depending on how good automation is, no good housing, no furniture, no medication, no clothing, etc, etc, etc.

You might want to research the physiological screening for those applying to winter over in Antartica, as well as submarine crew physiological screening. Also, you might want to consider not too much automation, so the crew has something to do (if you can't use suspended animation).

What do you mean by "small spacecraft"? Could you provide dimensions?

Not even going to space, 6 people living in some 20m2 flat go mad faster than 6 people living in 200m2. So dimensions matter a lot, and there's no mention about it except "small", which is very relative.