my apple tv 4k has no optical and my receiver has no hdmi inputs. what can i do

Tv does have a HDMI. But I run my sound thru an amp surround receiver. It has optical but no HDMI inputs.

Your TV may very well have the capability to passthrough the audio over optical through settings, instead of using TV speakers.

Apple TV →[HDMI]→ TV →[optical]→ receiver

Possibly you may need to change the Apple TV Audio Format (to DD 5.1 or Stereo), as optical doesn’t support PCM Multichannel (the Auto setting).

Try searching Ebay and Amazon for HDMI to optical (Toslink) adapters. However it may be easier to replace the receiver.

I did help a friend with an old Bose hifi get it connected using an old Apple Airport Express with RCA (analogue) output but they are not easy to buy.

An adapter is a very weak solution.

What you can do is (as long as you do not want change to a new A/V receiver) is as follows:

Connect Apple TV ----> HDMI Cable ---> TV Screen directly (this gets the video onto your screen)

Now for Audio..., what I would do is:

1- Get an Apple Airport Express wifi hub (like 50$), it has a very good quality 3.5MM audio output.

2- Connect the Airport Express to your A/V receiver (you need a cable one side 3.5MM and otters side whatever your A.V receiver takes.

3- Enter the Airport Express int your wifi.,, same wifi as the Apple TV.

4- Now when steaming movie or music on your Apple TV, choose the audio output to be the Airport Express.

I think this is the best solution for best audio quality.

I bought one of these in 2015 and it worked well:

https://www.amazon.com/ViewHD-Extractor-Support-Converter-VHD-H2HARC/dp/B00LKF6CPG/ref=cm_cr_pr_product_top

I replaced the (Toslink/Optical Cable) Hifi with Homepods last year and so no longer need it..

Please upgrade your receiver, it must be at least 15 years old.

The type of adapter that I recommended above costs around $30 and so is much cheaper than an Airport Express (if you can find one) or replacing the hifi or TV.

I forgot that it had RCA output as well as Toslink/optical and so I might use it to connect my friend's ATV to the old Bose hifi instead of the Airport Express.

There is no solution to this.

Your TV or receiver MUST have an HDMI input to you can watch Apple TV on your screen.

HDMI cable carries video and audio.