This is what I get if I try to open the video URL in a new tab:

Why the url is different? Have you tried the same video (https://assets.stuff.co.nz/video/production/1451429022625-darts best.mp4)?

I get the same thing with that URL too.

Occasionally stuff.co.nz will put up embedded Youtube videos, and they work ok.

Does that tell you anything?

Just that Opera can't open the video file.
What happen if you try to open the link of the video in a new tab?

I forgot to mention that the videos play ok in IE 11.

This is what I get if I try to open the video URL in a new tab:

There might be something changed in opera://flags/ You can try reset all options to default values. There's a button for that in the right upper corner.
Also, you can try disabling all the browser add-ons/extensions, such as ad-blockers etc.

I tried resetting the flags back to default, but that didn't help. I also tried turning off my ad-blocker but that too didn't help.

I suppose my netbook might be missing a video codec, but the videos play ok in Chrome so I don't think it would be a missing codec.

Chrome doesn't rely on the OS to play videos, it uses its own library afaik.
What does opera://media-internals/ says when you try to load the video?

Wow, I didn't know about opera://media-internals, thanks!

Under the Player tab it says:

render_id: 248
player_id: 0
pipeline_state: kStopped
event: WEBMEDIAPLAYER_CREATED
url: https://assets.stuff.co.nz/video/production/1451429022625-darts best.mp4
total_bytes: 19124866
streaming: false
single_origin: true
passed_cors_access_check: false
range_header_supported: true
error: FFmpegDemuxer: open context failed
pipeline_error: demuxer: could not open

Does that tell you anything?

Leo, he said he tried Turbo.
I guess in these "new and shiny" Windowses some mediapacks are missing. Or at least...
Try comparing those packs and stuff on both systems. Perhaps you might need to find that missing gear somehow...
You said "won't play": what does it show instead? Do you have an active frame/object there? Is there any context menu available in that area? Does it say something? Same stuff or varies?

This is what I see when the page with a video loads, without me having clicked on anything to make the video play:

I suppose my netbook might be missing a video codec, but the videos play ok in Chrome so I don't think it would be a missing codec.

Your using and out of date version of Opera time to update
latest version is 34.0.2036.25
and all the videos on Stuff work fine for me here in Napier

Yeah, I'm now running 34.0.2036.25, it updated since I originally posted at the beginning of December.

It's only one site that I've found that it won't play videos on, and that's stuff.co.nz

Do you have turbo or/and "click to play" enabled by chance?

Turbo is turned off. Click to Play is also off (it's set to "Run all plug-in content (recommended)").

Works fine here. Have you disabled encrypted media extensions in opera://flags/ ?

I tried that, it made no difference (but thanks for the suggestion).

I tried Opera on my desktop PC and the videos work fine on there, so it must be something to do with my settings on my netbook.

They use the same OS? Opera relies on the OS to play some videos.

No, the one that plays videos ok is still running Windows XP, the one that won't play videos is running Windows 7. It's only one site that I've found that it won't play videos on, and that's stuff.co.nz

I tried Opera on my desktop PC and the videos work fine on there, so it must be something to do with my settings on my netbook.

If I reinstall Opera will it keep all my bookmarks, passwords and form-filling entries?