I noticed that most jet engines are made out of aluminum or metal components. However, carbon fiber has a melting point of 3500 degrees celcius, way above metal or aluminum. So, can I build an entire jet engine out of carbon fiber?
No, you can not. Carbon fibre composites always contain a matrix component which serves different functionalities, but (in a micro nutshell) bonds and keeps the fibres in place. What you get is a very complex material with lots of lightweight potential, but also very different property profiles compared to metals.
Most matrix materials (thermoset, thermoplastics) will not work up to engine (combustion chamber) temperatures and either degrade/burn or melt/vaporize/burn long before reaching even a meagre few hundred Kelvin (say, 500K). In fact, carbon fibres will not work as well in an oxygenated high-temperature environment, and degrade/burn up as well.
One composite material that works and is used already in engines are ceramic matrix composites, like (amongst others) e.g. C/C-SiC structures, but those are very special animals (and even there the carbon fibre parts are susceptible to burn). Especially in terms of mechanical properties, these cannot compete with superalloys.
But also for many other, ambient temperature parts, metals are much better suited. Think of example of a ball bearing - doing that completely in composite will not work really well (if at all).
What is relevant is not the melting point, but the temperature at which the material loses its mechanical properties (such as stiffness or elasticity).
The epoxy (the material keeping together the fibers) will usually degrade much sooner than the fibers, basically voiding the benefit of the material.
There are studies to increase the resistance to high temperatures, but I have no idea how near we are to see them applied to critical components such as the engines.
Also note that usually the blades of the turbine and the combustion chambers are made of alloys designed specifically to resist the high temperatures in those areas of the engines.
When I was beginning my university studies in engineering 15 years ago our professor told us that they actually think of building jet engines of that material!
Carbon-based materials seem to be the perfect material for jet engines in nearly all points (heat resistance, mechanical strength at high temperatures, weight, ...) with one important exception:
These materials burn when they get in contact with hot air!
The professor told us that a lot of effort and money is spent in search of a coat or varnish that would protect carbon-based parts from the hot air. He told us that jet engines would be built of that material if such a coat is found.
He also told us that up to then no suitable coat was found - obviously this has not changed in the last 15 years.