For the events I want to run I would like my players to be able to do things that they can't do in the real world.
The players are reacting to a story. Even in a heavily Player versus Monster game you're relying on your players to interact and get involved with the world you've given them to make the story. Intense moments can come from putting players under stress and making them have to work to achieve things in difficult situations This tends to be a large part of games we run. Examples include the basics such as trying to rescue an injured comrade while people are trying to shoot you, and clearing a building knowing there's a good chance something bad is hiding in one of the rooms through to trying to replace parts for the power system whilst lying in a tunnel that's barely wider than you are in the dark with only one arm and a glow stick, or balancing meds on a critical patient and fighting to keep them alive.
Completing tasks can be a big component of a game. Strangely it's a big component of running games as well, except then the money real and I don't have all the parts.
Rules can help with this. Leaving things freeform is an option. It has several potential drawbacks which means whilst I would use it for some situations within a game, I wouldn't want to use it for everything. One of the first things I'd consider is whether or not it's something the players can represent without help, or whether it would be improved with help. If you have a player playing a mechanic as long as they've got some tools (either their own or ones you've provided) they can roleplay fixing things. If someone is playing a doctor they can roleplay trying to keep someone alive. They're things you can tell people they can do and then let them do them.
We like to give them puzzles. Sometimes they're just props to roleplay with and sometimes there are tasks with success/fail measures built in. For example we've given players piles of plumbing pipe and labelled it up. The players know (from a manual, or an instruction from a base computer) that they need water to flow from the inlet through pump a and b but not pump c, and the players can run with it. This can range from impossible without them finding something that's missing to ridiculously easy, but this is just something to help them roleplay their task. It gives them something to do that fits what they imagine the task should look like.
For the Alone games we have used various computer games to simulate tasks. This was appropriate for the effect we were aiming for (Alien based, so 1980s looking computing). It allowed us to give players skill levels that automatically applied to their logins. Doctors had higher medical skill sets so had longer to complete tasks. If the player was trying to do something they couldn't do the skill required for the game made the chances of them succeeding much slimmer. Giving up that control is scary. You're allowing players to fail at tasks and when it's something like medical care you could end up with a significant number of character deaths.
We wrote a scientific research game that involved the players finding slides, viewing them with a slide projector to find a number, and inputting this into the software. Some of the numbers on slides would give them access to files that told them some of the back story. Others wouldn't, or you'd need combinations to work things out. The slides were hidden across a 3 mile tunnel complex and there were 5 opposing teams of players.
Piloting can be done using systems like Artemis, or by giving a single player a computer terminal and a piloting game. Both of these allow a variety of outcomes, and success and failure. Roleplay is built in. However, my favourite piloting simulations were both from ZapFest. for the first We gave them cardboard consoles in a mocked up bridge and held up signs to tell them what was happening ("lurch left" "incoming transmission" "you're being hailed" etc). They reacted. It was amazing. It just worked.
We also gave them a darkened hall lit only with UV, and little spaceships and missiles on sticks. Players and Monsters dressed in black enacted spaceship battles through the space. It worked surprisingly well.
Going back to Alone, small tasks keep the players awake and moving. If the air filters have to be changed every hour someone has to be up to change them, and then they have to crawl through a face hugger infested tunnel to get to the space the filter is in and bring the spare filter with them.
It helps that we have a large workshop, and people who can program, build, and create electronic systems for us. Every event we rewrite the computer systems we have to make them work better and to be more future proof, and despite the hours of programming time we've not managed to reuse them yet. Once we've finished an event we stop thinking about it and the team changes and hardware changes we make for the next one mean that we start again. To make this something we can build on and share with other people would be useful, but isn't high enough up our priority list.