Is quantum mechanics applicable to only small things?

Might be that when the professor was telling you the answer he was expecting in lecture, you weren't paying attention, and now you're just looking for justification. Sure QM describes rules at the most foundational level of our comprehension, but those rules are most useful when applied to a certain problem domain, and the macro world isn't usually part of that (black holes aside). Though I would argue that "small" is a very simplistic answer - perhaps the most simplistic answer your professor would accept, where perhaps more specific answers would be preferred, not less specific ones.

Quantum mechanics (QM) doesn't deal with "everything". Otherwise it would be called The Theory Of Everything.

The most important obstacle is that QM doesn't deal with gravity. And since gravity becomes really relevant at large scales (with the exception of the Planck length) then there is some truth to your lecturer's judgement.

You are right in your understanding. Your professor is wrong. As mentioned earlier by others, quantum mechanics is applicable to the macroscopic regime too, but how to interpret the equations is something non-trivial and there is active research is pursued by many in this direction. Always remember, classical objects are quantum objects too.

If you are interested in knowing more, one such research area is called as macroscopic quantum mechanics (not something pioneered by Dr. Rovelli, but by Dr. Ravi Gomatam). Some of his papers are freely available from his ResearchGate account.

Or to get started, just go through his presentation here.

Your lecturer is wrong. Quantum mechanics would give accurate predictions when applied to macroscopic objects. The idea that quantum mechanics doesn't apply to macroscopic objects doesn't make any sense. Quantum mechanics explains the behaviour and interactions of atoms, and objects are made of atoms, so either quantum mechanics explains the behaviour of macroscopic objects, or it is false. The reason we don't see quantum interference for objects like human beings, pens etc has nothing to do with quantum mechanics not applying to those objects. Rather, quantum mechanics explains that when information is copied out of a system during an interference experiment interference is suppressed:

https://arxiv.org/abs/quant-ph/0703160

Since information about the location of large objects spreads into the environment on timescales that are a lot smaller than the timescales over which we see those systems evolve, those systems don't undergo interference.

Everything in the universe is such a broad word.

One thing that QM does not deal with is for example gravity. There are attempts to apply QM on gravity, but they are not successful so far and as it stands, QM cannot be applied here.

There is also problem with applying QM to everything at once. QM is quite problematic when it comes to explaining measurement. The standard formulation of QM introduces special agent to deal with it. So you need something outside of your QM system to act as this agent, which contradicts your attempt to apply QM on everything.

You may say, that QM should apply to everything as it is according to our understanding most fundamental theory we have, but that does not mean it does. Existence of quantum gravity might look promising, but we do not know yet. The measurement problem is however quite different and there is less hope it will be solved withing the framework of QM. It can be dodged as long as you retain some external agent - which is the strategy physicists adopted - but as long as you want to include everything there arises a problem. I think (I heard Lee Smolin to talk about it somewhere) research in quantum cosmology faces just this problem.

Edit

I would like to explain better the use of my word "agent". The problem is, that somewhere in the transition from QM to classical, the system must make choice about its state. The problem is QM does not define when does this happens, only how does this happens. It is up to the physicist to know when to apply the collapse during calculation, QM itself does not dictate this. The collapse itself is integral part of the QM, but when it happens is not. This missing knowledge that is left upon the physicist making the calculation makes QM not self contained and therefore it cannot be applied on "everything" in this sense. The choice must be made outside of its realm.

But of course this is based on standard formulation of QM I was taught. I do not follow research on this topic, so if there is more knowledge about this problematic, I would be glad to be corrected and read more about this. However, I remember from book by Sabin Hossenfelder "Lost in Math" that measurement problem is still huge hole in QM.

An example of (very) big things that need quantum mechanics to be properly described is black holes.

Is Quantum Mechanics applicable to only small things?

No. It's applicable to things which can be described by quantum numbers such as - spin, parity, magnetic moment, charm, x-charge, helicity and others. Also such objects are subject to measurement of entanglement degree if any. And to them applies uncertainty principle and wavefunction. Usual boundary helping to consider such objects are De Broglie wavelength. For QM objects De Broglie wavelength must be much greater than Plank length : $$ lambda_B ={frac {h}{mv}} gg L_{Plank} $$ For example for human of 70 kg mass, taking unit speed, gives De Broglie wavelength on the order of Plank length, so certaintly QM effects on walking human can be safely discarded.

Above given equation can be rewritten in terms of object volume :

$$ lambda_B ={frac {h}{rho~V~v}} $$

This gives insight that De Broglie wavelength can be comparable between objects of high-density/low-volume AND low-density/high-volume. Latter corresponds to Bose-Einsten Condensate,- a specific ultracold gas type where all gas particles are entangled together and because of that whole gas cloud acts like "one big quantum particle". I.e. BEC gas cloud is a macroscopic quantum mechanical object to whom all QM rules apply.