With Linux, you could use kexec jump to transfer control completely to another kernel (ie, your program). Of course, with great power comes great responsibility - it is entirely up to you to service interrupts, and avoid corrupting the old kernel's memory. You'll end up having to write your own OS kernel to do this. Also, the transfer of control takes quite some time, as the kernel has to de-initialize all hardware, then reinitialize it when it's time to resume. Since kexec jump was originally designed for hibernation support, this isn't a problem in its original context, but depending on what you're doing, it might be a problem.
You may want to consider instead working within the framework given to you by the OS - just write a normal driver for whatever you're doing.
Finally, one more option would be using the linux Real-Time patchset. This lets you assign static priorities to everything, even interrupt handlers; by running a process with higher priority than anything else, you could suspend /nearly/ everything - the system will still service a small stub for interrupts, as well as certain interrupts that can't be deferred, like timing interrupts, but for the most part the heavy work will be deferred until you relinquish control of the CPU.
Note that the RT patchset won't stop virtual memory and the like - mlockall will prevent page faults on valid pages though, if that's enough for you.
Also, keep in mind that whatever you do, the system BIOS can still cause SMM traps, which cannot be disabled, except by motherboard-model-specific methods.