Backup Exec 2010 with the Intelligent disaster recovery option configured and running would be able to do a full server backup without interrupting any services to tape or disk, which could then be used to do a "Bare Metal" recovery - Start with a blank server, and recover the whole system up to the point where the backup was made.
The two drawbacks in the specific case you're talking about is the cost of backup Exec (You'd need the standard software, the exchange agent, the oracle agent, and possibly others like the SQL and active directory agent) - From the sounds of it, they may not be willing to pay the licensing costs.
The other issue is IDR relies on having similar (not identical) hardware: if you have a serious fault on this hardware, you may have issues getting something close enough to put your IDR on in a hurry, especially given the age.
I had a similar issue with another client: They had an SQL 2000 based application that needed to keep running, even though the decade old hardware it was on was drying. We ended up virtualising the Operating System, and running it on a new Windows 2008 R2 Server using Hyper V - You said they don't wanna spend on the IT, but if you can get a low end server (or if really desperate, a high end workstation) out of your budgets, it may be the best all-round option.
Hyper V (And pretty much every Virtual OS) includes snapshotting so you can get a full, restorable backup point every night if you wanted, and what was a high end server 10 years ago that cost £10,000 -ish can now be done running on an entry level server: The customers system is running on a Sub £1000 Dell T100 server, with a noticeable performance improvement over the old hardware. If you really wanna trim costs, you could use the free Microsoft Hyper-V Server 2012 as the host OS (http://www.microsoft.com/en-us/server-cloud/hyper-v-server/default.aspx)
Know it's not the answer you wanted to hear, but those are the best options I could come up with for you, keeping costs as low as possible.
Good Luck with any forthcoming battles with the Finance department.