Ive heard that an integrated chips contains transistors, capacitors and resistors. However, can resistors or capacitors get that small to be placed on a silicon chips? Or is it just the arrangement of transistors that can give provide the functionality of any resistor/capacitor?
A capacitor is a structure of two conductors isolated by some non-conductive material, that can easily be made on a chip.
Capacitors on a chip are limited in value though as large capacitors would make a chip too large and therefore expensive. In IC manufacturing processes I designed circuits you can get metal-to-metal capacitors of around 1 to 5 femto Farad per \$ \mu m^2\$. That limits capacitors to around 10 pF to 100 pF for most purposes. Yeah, that's quite small!
Resistors can be made by any material that does not superconduct (as that would mean 0 ohms) or is not an insulator. There are plenty of layers suitable for this on a chip.
Resistors on chip can be made quite easily using metal tracks (for low values, like 0.01 ohm to 100 ohms). Or using some diffusion layer like the layers that are used to make the source and drain of MOS transistors. For most circuits however we tend to use polysilicon resistors, that polysilicon is the same material used for the gates of MOS transistors. Using these we can easily make almost any value between 10 ohms and 1 Meg ohms. High value resistors tend to be large in size though so we only use that when there's a good reason to do so.
So no, resistors and capacitors do not have to be made using transistors only. However sometimes transistors are sometimes (mis)used as resistors and capacitors depending on what component properties we designers want.
Resistors and capacitors are not made and "placed" on a IC (as a literal interpretation of the question would imply), they are manufactured as one monolithic process, the same way as the transistors are made.
The capacitors are created of metal areas separated by silicon oxide dielectric. With 1mm2 area and metal layer distance of 0.1um the value of capacitor is about 100 pF (see this researchgate answer). However, the silicon area is expensive, and areas of 1mm2 are not common. Multilayer integrated caps are also possible, making the capacitance bigger.
Integrated resistors are made by various levels of ion implantation of pure silicon areas (see P.13 of this lecture), and can be of nearly any value, although they are not very powerful electrically.
ICs can also contain inductive elements as well.
All basic ideas of silicon manufacturing are widely explained on the web, Wikipedia has a good introductory article.
There's something called sheet resistance, long story short is that you can make many different sizes of resistance while only focusing on the length to the width ratio. Depending on the material and thickness of the material, you will get different constants for your "per square" value.
So let's say that your resistance per square is 1 Ω, then if you connect two components with something that is 1 µm wide and 2 µm long, then you have a rectangle made up of two squares, and the resistance will be 2 Ω. If you instead have a track that is 100 µm wide and 200 µm long, then you again will have 2 Ω of resistance. Do you see here that you can make all the different resistances regardless of the size? If you for some reason cannot make your resistance, then you can make a resistance multiplier with a operational amplifier.
Capacitors can always be the gate of a MOSFET, since the gate of every MOSFET has capacitance. But it doesn't have to be the gate of a MOSFET, it can just be two plates near each other. This value can, just like a resistor, also be multiplied to absurd values, by using a miller capacitance multiplier that also uses an operational amplifier. You don't have to use operational amplifiers for everything, you can also just use BJTs or some other tricks you might have up your sleeve.
Keep in mind that the energy stored is not "absurdly" more, you're essentially changing the impedance so the circuit behaves as if there would be more capacitance. It's similar but not exact to say that I have 1 dollar, but some other part of the circuit speaks in Yen and "sees" 111 "Money" (in yen). Maybe it's a bad metaphor.
MOS transistors inherently provide capacitors, using the gate/bulk capacitance.
metallization has resistance, about 50 squares per ohm. The gate poly often is quite high sheet-resistance. Again, modern MOS processes provide many R + C construction methods.