Let's say that I just replaced my engine oil but decided I want to lift the valve covers to do some gasket replacement. I know it may sound cheap but I'd rather not throw away $25 worth of almost new oil. Would it be safe to catch the old oil in a receptacle and set it aside and put it back into the engine once I am done with my repair?
As @dodgethesteamroller stated, there is nothing wrong with reusing the engine oil. There are two caveats and one concern I'd like to mention with this.
First, use a clean container to store the oil in while you are doing your engine work. Your normal engine drain pan will probably not give you the cleanliness you are looking for. You DO NOT want to have the dirt and crud which collects in the pan to be put into your engine.
Second, it's probably a good idea to change out the oil filter when you are doing this. It doesn't hurt anything and should only cost you a few dollars to replace.
Concern: Before you decide you are going to reuse your nearly new engine oil, you need to realize the reasons for doing the work on the engine in the first place. If you are replacing worn bearings, you don't want to reuse your engine oil no matter how new it is, because it will more than likely have metal fragments from the old bearings in it ... no sense in putting metal particles back into your engine and having all those wear items thrown into the mix. It just does not make sense.
I don't blame you for wanting to save a buck and/or the environment, but don't ruin your new engine work in the process by not utilizing some common sense.
Nothing wrong with doing this at all. Just make sure that you keep the oil clean and put back in as much as you take out!
Usually you don't think about keeping oil clean as it drains since you're going to dispose of it anyway, so take precautions: use only new or thoroughly cleaned funnels and drain pans, wear clean rubber gloves as you unscrew the drain plug, and clean the surfaces of the engine below and around the plug as well as you can beforehand, if your car has a tendency (like most do) to dribble some oil partway down the block or along the pan when you drain it.
It would be a good idea to keep the "almost-new" oil in an airtight container with as little air space above the level of the oil itself if it's going to be out of the engine for more than a day or so, to minimize absorption of moisture from the air. (Don't worry too much, as a little water in the crankcase does no harm--it's generated all the time as a byproduct of combustion and normally just boils off when the engine gets hot.)
While new oil is always preferable it is entirely possible to drain the old engine oil out and run it through some fine filters (even like one or two of those paper paint strainers or an old cotton t-shirt and then use it to fill to back to the proper level. Old oil is better than no or to little oil and better than old oil is new. Just my opinion anyway. Hope it helps