What's the best way to blacken metal jewellery to this quality?
A reverse image search suggests that the item of jewellery in that picture is plated in black gold; an alloy made of cobalt and gold. This might be ...difficult to do at home. Let's look at some alternatives.

Painting

This is probably not what you're looking for, but good quality matt black spray paint, sprayed over a decent metal primer coat, will be reasonably durable and very simple at achieve

Oxidating (Bluing)

You can either apply a coating of an oxidating substance, or oxidate the metal itself - a process known as bluing. This will give a relatively matt finish, but probably not as matt as you're looking for.

Anodising

If you go for aluminium instead of steel/brass, then you can try anodising it. Anodising at home is tricky, and the cost of tools is probably more than it's worth if you only want to blacken one or two items, but it is possible.

It seems to be possible to get a matt finish when anodising at home

Powder Coating

Powder coating is very durable, and can be done in many colours, and is doable at home. Like anodising, it requires some relatively specialised (and expensive) gear.

Outsourcing

This is the boring option, but if you need something doing well and don't know how to do it, or don't have the necessary equipment, it's often possible to pay someone else to do it for you.