If you use PC hardware for your servers, then a so-called “KVM switch” is one possible solution. A KVM switch allows the use of a single keyboard, video screen and mouse for multiple boxes. This cuts down on the space problem, but only works for PC hardware (not any communications gear you might have), and is not accessible from outside the computer room. Nor does it have much scroll-back or logging, and you have to handle alerting some other way. The big downside is that it will not work for serial-only devices, such as communications hardware. This means that even with a room full of PC-based servers, you are probably still going to need some sort of serial console solution.
Note: Actually, Doug Schache has pointed out that you can get KVM switches that also do serial consoles or Sun compatible KVM switching as well as PCs, but they are expensive. See Avocent for example.)
You might be tempted to do without a console terminal, but when things go pear-shaped you really need to see what is on the console. And you have to use the console to boot the machine and do things like OS upgrades or installs.
You might try having a single console terminal and switching from server to server as needed, either with a serial switch or just by patching it into the required machine. Serial switches are also hard to come by and not cheap, and may cause problems with sending BREAK when they switch. And (if your computer room is anything like ours) you never seem to have the right combination of patch leads to connect to the machine you need to, and even if the leads are there you can never work out exactly which combination of DTE/DCE headshells goes with which lead goes with which hardware. So you spend the first 10 minutes fooling around with breakout boxes and a box of leads, all while the server is down and the users are screaming. Of course this does not deal with the logging or remote access requirements. And inevitably the console is not switched to the machine you need so you lose all the console messages that might tell you what is going on.
One popular solution is to use terminal server hardware. Typically, the serial ports are connected to the various machine consoles, and set up for “reverse telnet” access. This means a user can telnet to a given IP/port and be connected to the appropriate console. This can be very cost-effective, as suitable old terminal servers can be picked up fairly cheaply (assuming you do not have a couple lying around). And it is of course network-accessible so suitable for remote access. But it suffers from one major drawback: if the network is down, then you have no access to any console, even if you are standing right next to the machine. (This may be partially alleviated by having a suitable terminal connected to one of the terminal server ports and connecting from there, but the terminal server software may not support that.) Also there is no logging or replay of console messages. But with a bit of work, and the addition of some software such as conserver (described below), this can be made to work pretty well.
A possibility suggested by Bron Gondwana is similar to the above solution. If you use servers with multiple serial ports, you can connect each spare serial port to the console port of the “next” server, creating a ring of console connections (in some sort of order). This can be made to work reasonably well with the aid of the conserver software, but can be a bit confusing otherwise (i.e. remembering which port is connected to which console). And you are stuck if you need to use serial ports for other things (such as modems) or you have machines without spare ports.
Or, if your budget exceeds your willingness to hack, you can buy an off-the-shelf solution. These vary in price and capability. See, for example, Lightwave, Perle, Avocent or Black Box. These solutions can be quite expensive - typically $USD100 - $USD400 per port.