Can cables and wires for home internet emit harmful radiation?

From Wikipedia:

Ionizing radiation is generated through nuclear reactions, nuclear decay, by very high temperature, or via acceleration of charged particles in electromagnetic fields. Natural sources include the sun, lightning and supernova explosions. Artificial sources include nuclear reactors, particle accelerators, and x-ray tubes.

(emphasis added)

Your wifi router or ethernet cables aren't going to emit any ionizing radiation.

Is there a certain minimum distance where the risk becomes almost nil? What if you are over a foot away?

There is no known risk at any distance due to electrical fields. There is absolutely zero risk due to cancer from these fields because we know how cancer is caused and it isn't from electrical fields.

If you built a house next to a radar transmitter you might have some trouble. Otherwise, you're fine.

There are known risks from high intensity electric fields. For example see the UK legislation "The Control of Electromagnetic Fields at Work Regulations 2016" at https://www.legislation.gov.uk/uksi/2016/588/pdfs/uksi_20160588_en.pdf

However these regulations apply to field strengths higher than 100 volts/meter (and at low frequencies, 10,000 volts/meter) which is much higher than anything you will find in a domestic environment.

This guide to the regulations https://www.hse.gov.uk/pubns/priced/hsg281.pdf gives examples of the type of devices where the issues need to be addressed. None of them are likely to be found in a normal domestic environment. See Table 2 on page 9 for a list of devices which will not exceed the limits of the directive.

But if you plan to live inside a airport radar transmitter or a hospital MRI scanner, for example, you might have something to worry about :)

Like others have stated, low intensity non-ionizing electromagnetic radiation (the radiation from wires in your home) has no known health effects. That said, it does not hurt to minimize EMI as it allows your electronics and wireless devices to operate more reliably.

Speaking specifically to Ethernet wires; these have one of the lowest radiation of all consumer grade wires because they use a twisted pair of wires that cancels out any emissions. They are designed this way to protect the signal on the wires from external noise, but this effect works both ways. To additionally reduce the emission you can buy shielded Cat5, Cat5e, Cat6, Ethernet cable (Shielded Twisted Pair). The foil around the wire forms a EMI shield further reducing any emissions in or out. And for further reduction you can add ferrite filter around the outside of the cable at the Ethernet switch to reduce "common mode" emission.

To put it into perceptive, the 60hz electrical wires in your home, and FM radio station transmitter on the other side of town have orders of magnitude larger intensities in your home than standard unshielded ethernet wires. Typically the largest EMF emitters in the home are low quality power supplies that are not FCC approved. The FCC sticker on the power supply indicates that the device design has been tested for EMI and complies with United States regulations. You can check the back of your router, modem, and both power supplies for this mark. Most likely they will all have it, but if not I recommend getting a different brand.