We're moving to cheap office space that provides a lot of stuff: receptionist, copiers, printers, microwave and internet access - all of it shared with strangers.
What should I ask to find out if the network is safe to use, and if it's not inherently safe, what can I do to make it safe?
In short: Be just as careful as you would be with any other public/shared internet connection!
Consider hooking up some decent firewall/router in between your own pc's and the building network (just as you would with an internet uplink). This way you can keep control of all traffic between your own machines.
Consider it a hostile network if you share it with strangers.
You can put your own section of the network behind a firewall, so the rest of the office is outside. You might have to have your own printer for that though.
To set up such a firewall, you would have to only use one of the office network connections, and run your own network connections to your firewall machine. Typically the network ports in the wall run back to a central room where they are patched into a router or something. If you do not trust this network, you can treat it as another part of the "external" network, and run your own internal network in your office.
Be warned though. If someone can physically get into your office, they can gain access to your computer, or use your network cable to connect a laptop and do things you would like to disallow. You might not want to store any information on your office computers, maybe use just a VNC session. Or you could use laptops and not leave them in the office. If you can physically lock your section of the office it would be a good idea.
Never connect to servers through this network unless you are on an encrypted connection. SSH tunnels should be used for anything that isn't normally encrypted.
There are a lot more things you can do to improve your security. Basically considering the network to be hostile, and assuming that someone is always eavesdropping would be the safest attitude.
A good hardware Firewall would be an excellent investment for the company, but might be a little rich for one person. Comodo firewall can be set to a paranoid level, even locking down the registry, and is free.
Install Wireshark on your computer, it will allow you to watch all of the network traffic on your computer, and on a second note will allow you to see how much chatter goes on and shut down unneeded services.
Shut down all unneeded services, every port that is open is one more place for attackers to slip through.
Use paraphrases with special char instead of passwords. It would even be better to have pass cards or fingerprint readers in addition to the paraphrases, not instead of.
Physically lock up all computer towers, switches, and routers in cabinets if possible.
Check the network and replace any hubs, if found, with switches. Hubs are BAD.
If scanners or copiers are on the network, make sure that they are covered with ACL's in the router. Attackers have been known to access networks via dial in modems on copiers.
Lastly, make sure there are no modems on your network.
Regarding the OP's comment to thijs's answer (I can't add a comment yet):
Talk to whoever's in charge of your new office space. The existing connections to the offices have to lead back to a patch-panel/router somewhere. You might be able to put your specific offices/equipment onto a separate network through their existing infrastructure.
Are you going to be assigned static desk/office space? Or is it more of a "hotel" arrangement where it's first come, first served?
If you are able to have a static (and secure) location, just stick your own firewalling router at the port. That will keep most of the problems out.
Edit to add:
I don't know how sophisticated your shared-office provider is, but you might want to ask if they can assign a virtual LAN (VLAN) group for you and your coworkers, and then put that VLAN behind a firewall. That'll be the easiest way to go.
Check to make sure they do not have any unsecure wireless networks. At my last office/job it wasn't that we shared space but the day that I decided to switch our office over to wireless routers I found that there was already a wireless lan 'open to the world' from the medical health facility right thru the wall in the office next to us. Given all of the HIPPA rules, etc, I promptly told them of THEIR problem. I documented the conversation with their office manager, as well as sending a written document to them which we filed in our records... Not so much because I'm a nice person but because I didn't want them coming back saying that WE had done something on their network. If their system is insecure you need to safeguard yourself against their ability to blame anything on you if something goes bad with them.
If you want to be really paranoid, you can install VPN clients on every computer, and have a off-site VPN server at a datacenter... But that eats into your bandwidth, and is probably more complexity than you want to put up with.