As I understand I can just purchase options that trade 100 stocks. Are there possibilities to buy, lets say, a put option to sell just 5 stocks at a given strike price?
Traditional exchange traded options are for 100 shares.
There are post reverse split contracts that cover less than 100 shares but because of the adjustments, the terms of the contract have changed. These are often complex and should be avoided if possible.
The CBOE introduced 1/2 a dozen or so mini options about 6-7 years ago but the public never took to it.
In a few cases (but not many) yes. The SPX mini for example.
You can buy any option contract you want if you can find someone willing to take the other side of it; this is entirely common in real estate, for example.
In the case of publicly traded stocks, however, option contracts are standardized specifically so that they can be reduced to a simple code (I happen to be involved in some
SFL 200515P00010000) and routed through automated trading and clearing systems. Based on experience, the institutions developing the standards decided that the standardized contract would be for 100 shares of the underlying, at certain prices (you usually can't trade an option at strike $11.21), expiring on certain days (Fridays, especially third Fridays). If you want to use public exchanges, it's take it or leave it. This dramatically reduces the transaction costs and thus improves liquidity.
(There are a few exceptions, such as when the contracts have to be "adjusted" for splits or similar, and "mini" options as @Philip mentioned, but in general, what's available is standard or nothing.)
The Options Clearing Corporation and the rest of the industry went through an initiative a few years ago to standardize options symbols (the Options Symbology Initiative or OSI) so that they could support mini options. These would be typically for 10 shares rather than 100, and many exchanges offered reduce fees for mini options over regular options... except Amex. It ran for a while but the volume didn't pick up, so it was abandoned.
Today, any option for less than 100 shares is usually called an 'adjusted series' which often results from some kind of corporate action in the underlying stock. Market makers are not obliged to quote them, so liquidity is poor.