The problem with student and any small move comes when a small load has to be transported a long way:- For the moving company, the mileage and road time costs are not much less than for moving a whole house...So your quote is unacceptably high.
There are two ways to approach a solution to high cost removals: You could hire a van, (and there are van hire companies that will rent a small van 21 year olds, sometimes one way), or you could look for a removal company with whom you can part load with someone else going the same way.
One way van hire could be particularly attractive if you could then find a partner to split the cost, and get a one way deal, but the problem of finding a qualifying driver may be insurmountable.
So the problem for most student movers is in finding a company who will part load at a good price.
1) Spend some time phoning around: There is an element of luck in finding a van going your way.
2) Be as flexible as possible with your date - maybe you can organise someone at your destination to receive your property.
3) Try to find your own partner who is also going your way, and approach a removal company with a joint proposition.
Apply common sense to make sure the job is easy to understand and execute for the remover
To get the best price it is vital to convince a remover that your assessment can be trusted.
4) Stop and consider the quantity of your removal:- because removals tend to be arranged before you have packed, peoples own estimates are often hazy, usually under assessed, and sometimes the actual job bears little relation to that described.
5) Package and box as much as possible, for speedy loading, protection, easy trolleying, and secure, tidy stacking on the van.
6) Consider any likely impediments to access or delivery (like locked barriers on campus)
7) Offer all the assurances needed about having your things easily accessible and that you will be ready to go before the van arrives.
The author has spent 25 years in various service industries, including the last 8 in the removals sector. His website http://www.student-movers-forum.com is a resource for all small movers. You can find sources of low age van hire, specialist part load removers, packaging, message boards and advice.