A hangar deck stores 20 HS worth of ship, and a boat bay stores 5. These connect infinitely, so you can load a ship with any size ship if you add enough hangar. One player developed a 700 box launcher platform carried via a mothership, for when surgical strikes just weren't enough! It needed to be taken inside the hangar to reload all the launchers. I've seen others who carried whole battleships in a hangar.
Ideally, for a dropship, you'll have a mothership that sits some ways off, outside of fire range. This mothership should have troop bays and hangar decks, and several cargo controllers (those ones that speed up loading and unloading). To deploy, you mount the marines/heavy assault into the shuttles, and launch the shuttles as a new task group. The shuttles then run to the target planet or ship and deploy, and then run back.
The shuttles should be as small as possible, with powerful engines (use that +5% thrust -10% efficiency type) preferably loaded with some type of fighter or gunboat engine for max thrust. Or if it's too big, fit regular engines and multiples of them. Play that part by ear. They should be outfitted with a combat drop module, a powerful engine, and little else. They need to be going 20x faster than the target ship to deploy soldiers with no casualties, but planets don't matter because they don't move. Still, you'll want speed so you can evade PDC defensive fire. Your drop shuttle cannot be too fast! Keep fuel low as well, try and measure a trip too and from your target and set your mothership that far away, so your shuttle isn't burdened by fuel or anything else. You're basically building a two-way suicide pod. Or you could make it a one-way shuttle and just give it enough fuel to make it over there.
Remember, if you drop a group of soldiers WITH their division/brigade HQ, they get a combat boost. You can deploy an HQ on a hostile planet or to a hostile ship, so you can fit soldiers and their HQ into an enemy ship.
A note: my screen's max display doesn't work, but the one right under max does. Different resolutions give different sizes, and a smaller size can give more vertical height. try lowering your resolution.