Sooner or later, there has to be a backlink. Whether that's from social, one of the "website worth" sites, or from DNS info sites.
Unless you "ping" the search engines (which I don't do, because it sends a signal that I'm looking for backlinks), won't find you. So, backlinks are important to at least get you started.
That's my caveat.
However, there are instances where I didn't tout for a single backlink, but concentrated more on internal linking to gain rank. It's more difficult today, but 10 years ago if you had a large website - ie more than a few hundred thousand pages - you could establish a good rank (and hold it) from anchor text within internal linking strategies.
I've done the same over the last couple of years to see if internal links still work. They do, but you won't get the same kind of traffic that we saw back then. But it's a strategy that can still work, if you play your cards right.
Other than that, there is very little you can do to rank a website without any link building strategies. You can get a foothold with a few links from high-DA websites, but link building has got to be part of a bigger, overall strategy if you want to pull in traffic from the SERPs.
Link building represents 80% what moves a website onto the first page of Google! (as long as your targeted page URL is utilizing The proper code to text and keyword density for your desired keyword)
The days of doing on page, building a couple citations or directories NAP and the occasional on-site blog (with an internal link) have been over for at least five years.
 furthermore any agencies that find success with the above prehistoric approach are normally targeting extremely small Geo areas and or super niche/longtail keywords!
Proper backlinking is what’s going to keep your clients paying you month after month!!
There's 2 major rules. If you want to rank for local low volume keywords you can probably do it without links. If you want to rank for any high traffic keywords it's almost a certainty you will need backlinks (and quite strong ones)
1) Not important if you can get a lot of high quality natural links
2) You can rank for keywords without building backlinks by creating content that nobody is searching for or content people are searching for that has extremely low keyword density. If this content ever becomes popular, you might get natural links.
3) How many what?
The primary way google determines E-A-T (Expertise, Authoritativeness, Trustworthiness) is backlinks. Assessing backlinks is the primary method Google (and Bing) uses determine the organic value of your site. If your plan is to have no backlinks, I wouldn't waste your time doing any SEO really. You might as well just use Google AdWords, facebook, youtube, or whatever platforms suit you.
Make great content and you don’t have to spend time building backlinks. Without valuable content to get you backlinks organically you are destined to struggle with authority or rankings. The rare exception would be for local seo where you dedicated yourself to citations in place of backlinks
Don't go crazy in making backlinks , focus on creating quality content .. as mentioned by someone in one of the above comments.. once your website started ranking you will eventually get backlinks via different mentions, social media channels, tools, etc . You need to make sure to analyse those backlinks to get advantage of that.. many times I have seen lots of spam backlinks created by some unethical folks ..so beware of that
I’m so frustrated with link building! I’ve tried reaching out personally, commenting, forum signatures but no real luck. I’ve even tried the most popular services on Legit and Fiver but the links are usually terrible.
For me its still important,and especially if you're looking to rank for a very competitive keyword. You can rank without links but don't expect todefeat your competitors without off-page