A IT admin ask me to run the following command on the server so he can access that computer remotely with ssh

ifconfig eth0 1.2.3.5 netmask 255.255.255.0 up; route add default gw 1.2.3.1

But how does this work? What is ip 1.2.3.5? Can he access my computer at 1.2.3.5 (there is a static ip for the server, which might be used internally)?

1.2.3.5 is simply a perfectly valid IP address - nothing special about it that I am aware of. However, it might be that access control lists on your LAN are set up so that SSH would be allowed for that address, but blocked for your server's usual IP address.

it is two commands:
1) ifconfig eth0 1.2.3.5 netmask 255.255.255.0 up
2) route add default gw 1.2.3.1

1) configures your system's interface eth0 with the ip 1.2.3.5/255.255.255.0
2) configures the default gw to use for routing outside of 1.2.3/24 network

IP 1.2.3.5 is the IP you assigned to your system's eth0 interface
Yes, access to your computer via 1.2.3.5 on the network is possible as long as firewall ports are open and service is listening and remote user/system can route to that IP and know a valid user/pass.

Well, if I change 1.2.3.5 to the static ip, it works and I can acces it via ssh from other place. But it seems fail for 1.2.3.5. Is it possible to have two ip binding to eth0?

What is it that is failing with 1.2.3.5?

Anyway, yes, it should be possible to define a sub-interface, e.g.:
There may be a problem with the routing rule - if you defined 1.2.3.1 as the default gateway, then the internet would no longer be reachable via your other ip address. You'd probably have to add a specific static route for the computer / network from which SSH access is needed, e.g.
... would add a route to the network 80.254.3.24/29 (address and mask picked at random), via 1.2.3.1. The machine should choose the most specific route when making routing decisions (so should route all traffic to that address range via 1.2.3.1, but all other traffic should still go via the default gateway).

Hope this helps,
Rob