All the "digital" timer modules of this ilk will retain their timing settings without power applied. The parameters are stored in non-volatile "flash" memory as used in all modern consumer electronics. There are some low-cost timer modules that have knob-adjustment (potentiometer) to set the delay times; obviously these too retain their time delays by "mechanical" memory.
Note that if trying to pair multiple "digital" timer modules, with the microcontroller technology used in the low cost modules, you will get excellent repeatability (to within a fraction of a percent)...but the absolute accuracy may be in the 1% range. In other words, set two low-cost "digital" modules to 240. minutes and they may be off by, say, 2 minutes.
I bring this up with regards to your proposed scenario of first one on is first one off after 240. minutes. You may need to fine-tune the modules. You may need to set a particular module to 239. minutes to effect a 240 minute delay. Or, you may need to set a different module to 241 minutes to effect a 240 minute delay. Again, the repeatability will be better than the accuracy so once you make this adjustment, you should be good to go. This is a your-mileage-may-vary and I'm sure there are digital modules with absolute accuracy in the fraction of a percent such that 240. minutes will be 240.0 minutes within a second or so.
The video was just a "proof-of-life" showing what it looks like when running and its relative size to O-gauge. The particular module I showed has a wide variety of "modes" as described in the eBay listing. I was not meaning to suggest the mode used in the video was applicable to your situation. I do believe once I fully understand your timing requirement that this module has a mode that would apply. I have spent some time trying to decipher the mangled English translations of the instructions for this and other similar modules. For example, in this purposely blurry diagram I tried to sketch out the different modes and how the trigger starts, restarts, or even affects the timing. Purposely blurry is because it's a work in progress that I think is correct but I learn something new just about every time I try it! If this module is something you want to pursue, I will confirm it would work in your application.
There are some modes that do NOT require a trigger at all. That is, just apply DC power and its relay will open/close with the settable timing parameters. Some modes are "re-triggerable" meaning if you apply a 2nd trigger while a timing operation is running, it will restart the timing thereby extending the over-all cycle. There are some modes where, rather than the application of the trigger it's the removal of the trigger that effects some behavior. And I've found on multiple occasion that the module may come with MORE modes than are documented in the eBay/Amazon/wherever listing description! And so on. Can get confusing. But to some, the allure of finding out what's really behind Door #2 is part of the fun! 
