No, you won't eliminate mistakes. But you might be able to catch the most important ones. It's important to be conscious of what you are doing and look for ways of doing it more safely.
Identify risky endeavours. You can't be on your guard every second of the day - but everyone makes tiny errors all day long. Most mistakes can be fixed effortlessly, so get better at noticing when something has the potential to go wrong in a way that can't be so easily fixed. Any communication with a client, any deployment to a site, any global change to a document, etc.
Question what you are doing. Do you understand what you are doing, why, and how? Can this be done more safely? Are you sure it is better to do it than not do it?
Pause before going ahead with risky endeavours. Don't send an email without checking recipients, subject, content, and attachments. One by one. Take your time. Sending a quote for Client A to Client B could lose both clients. Better to spend a minute or so checking it through. Don't be tempted to rush, even in an 'emergency'.
Have a recovery plan. Know in advance what you will do if something goes wrong. Prepare so that if you do need to rectify your mistake, you can do it quickly - but without rushing. How will you know if you did something wrong? Is there a way to find out sooner?
Document anything that has many steps. If you have documentation, follow it like a checklist so you can tell if it's good enough or if it's outdated, incomplete, or faulty. Even if you only do it once - before you write your report, write a list of all the things that need to go in.
Track to-dos especially if forgetting is a problem. Anything you need to do and aren't doing right now should go in there. Track them all in once place and keep going back to your to-do list. Don't rely on post-it notes folded up and stashed in your back pocket.
Automate wherever possible. If you have a list of five things to type in, can you write a programme to do those five things and then you only have one thing to get wrong? If you are making software, can you run automated tests to pick up when you introduce bugs? Do you aim for 100% coverage? Do you test with the goal of breaking your code? Do you have continuous integration so you don't have to worry about forgetting to run tests?
Ask for a second opinion if you are doing something risky that you can't verify and can't easily recover from. It's a common practice for software that code is always tested by someone else before being accepted.
Learn. Every mistake made is an opportunity to revise the process.
Take care of yourself. Find time during the day to clear your mind and have a few minutes of silence or relaxation. Before work, after work, lunch breaks, etc. Get enough sleep. Eat regularly and healthily.