I have an old archived site with tons of text content. I did a google search, and the text does not exist anymore in the search and it shows as results not found, this is after using the " " operators.
Can I use that text and put in on my new website (new URL) or will Google see it as duplicate content?
I use content from my older sites offline many times , I just had to wait for the old pages to de index , had no issues. Sometimes I had to request pages removed but patience is best.
Yes, you can use it, but only when content is created by you. Not copied from Web Archive, or any site that block search engine spider.
It's not about the content exist in search database or not, it is all about, the content is your own. May be someday, the original content creator find out, you have copied their work, then he/she has right reserved to take legal action against you.
There are two cases :
1) Your old site is still active: as in the domain is still with you, your site is still hosted
In above case you should 301 redirect the page with the a content to similar content on new site
2) In case you want both to exist but index new site with crawlers, make sure you add a no index meta on your old site.
If you make sure the old content is deprecated either via 301 redirect or no index or 410 permanent discontinue, google will not penalize you.
Yes if that content is fresh and website is not working anymore, you can use it. But still I will advise you to not use exactly same content, it better to change the content little bit
