Quick answer: Paul is not saying that no one is righteous here, he is explaining to the Romans that Jews also sin and are not sin free just like all people, and he quotes King David's writing where he is talking about the fools and the workers of iniquity, King David is not talking about everyone but just those bad people, he is saying none of those bad people are righteous. So by Paul pointing that example out he is showing how Jews as well are under sin.
False teachers may quote Paul at Romans 3:10 and say that none are righteous.
They start using this in a way that contradicts nearly the entire Bible, and the bible actually calls many people righteous.
This is a poisonous teaching because it makes people believe they can never be righteous no matter what they do, as in there is no point in doing good works.
So these people who are convinced that they can not be good at all, don't have much motivation to do anything that is said to be righteous in the bible, and the only action they may do is say with words that they accept Yahushua the Messiah, which taking part in the covenant through his blood is only one part of being righteous since his sacrifice for us pardons our former sins and makes us perfect and righteous in the sight of YHWH.
But if you're not practicing righteousness at all then you no longer have pardoning.
Let me start off with a verse that will quickly show this misconception is not right;
This verse is talking about Yahushua the Messiah when it says "he";
1 John 3:7 "Little children, let no one deceive you, Whoever practices righteousness is righteous even as he is righteous"
So, let's strive to practice righteousness, and the more we practice it, the better and more mature we become in it.
Back to the misconception, the false teachers are reading Paul's words out of context and twisting what Paul said, not only that but they misunderstand the source of what Paul was quoting.
Here is the verse;
Romans 3:10 "What then? Are we Jews any better off? No, not at all. For we have already charged that all, both Jews and Gentiles are under sin, as it is written, None is righteous, no, not one"
Paul in this letter was simply pointing out to the people he was speaking to who were Romans, that Jews were not sin free, or exempt from sin.
At Romans 3:1 he had just pointed out there was a benefit to being a Jew, and then he was pointing out though they were special in some ways, they too were susceptible to sin just like everyone else.
This was an important thing to point out to the gentiles, because the pharisees in those days were likely teaching things that would confuse them from the truth.
Real quick I will show the verse before where he is talking about "Jews" having a benefit;
Romans 3:1 "What advantage then is there in being a Jew, or what value is there in circumcision? Much in every way! First of all, they have been entrusted with the very words of Elohim"
So with that being said let's look at all of what Paul said, which if someone bothered to read beyond Romans 3:10 would probably realize they were twisting his meaning;
Romans 3:9 "What then? Are we Jews any better off? No not at all. For we have already charged that all, both Jews and Gentiles are under sin. As it is written, None is righteous, no, not one, no one understands, no one seeks Elohim, All have turned aside, together they have become worthless, no one does good, not even one, Their throat is an open grave, they use their tongues to deceive, The venom of asps is under their lips, Their mouth is full of curses and bitterness, Their feet are swift to shed blood, in their paths are ruin and misery, and the way of peace they have not known, There is no fear of Elohim before their eyes"
He says 'we' as in we the Jews and quotes a well known Jewish man from the Bible...more specifically who YHWH himself says that this man is righteous and sought after YAH with all of his being. That man was King David, who did slip up and sin but was still called righteous.
First before I go to David's actual scripture, read what Paul said and understand what false teachers are teaching.
Is it true that none is considered righteous?
No one understands?
No one has ever sought YHWH?
Every man has turned aside and together they have become worthless?
No one does good not even one single person?
Every single person's throat is an open grave that they use their tongues to deceive and their mouth is full of poison?
Their mouths are full of curses and bitterness?
Every single person's feet are swift to murder?
In every single persons path they leave ruin and misery?
Not a single person has known the way of peace?
Not a single person has ever feared YHWH the great and awesome Elohim?
Good thing the false teachers who are teaching this doesn't even believe it themselves.. because the world would be scary if they gave up on trying to be good because they believed none could do it and the only way to be righteous is by the Messiah being righteous in us just because we say with our lips that we believe in him, and believe we are allowed to continue being lying, murdering, lawless devils because we can't help it.
Paul himself said if you sin intentionally you're lost, he said if you truly do not want to sin but you stumble and sin anyway, then you're okay because you're struggling with the sin in you.
The people who use this false teaching testify against themselves that they do not know the word, because if they actually read the word they would know Paul was quoting David.
Not only that but false teachers lead people to believe the "old testament" is useless to a "Christian" and none of them even care about it and treat the scripture from it as something that has become void. If that is true then why does nearly everything Paul say come straight from the "old testament"?
Moving on to the actual scripture from David that Paul was quoting;
Psalm 14:1 "The fool has said in his heart, There is no Elohim, They are corrupt, they have done abominable works, there is none that does good. YHWH looked down from the heavens on the children of men, to see if there were any that did understand, and seek Elohim, They are all gone aside, they are all together become filthy, there is none that does good, no, not one. Have all the workers of iniquity no knowledge? Who eat up my people as they eat bread, and call not on YHWH. There were they in great fear, for Elohim is in the generation of the righteous"
David was talking about FOOLS and the WORKERS OF INIQUITY if you read everything he says and in context, he was not talking about everyone. Notice he says the fools and then says "they" are corrupt, "they" have done abominable works, "they" have all gone aside.
After David says this about the FOOLS, he then says that YHWH is with the RIGHTEOUS.
That contradicts the false teachings!
YHWH was with David and many other people, even before Messiah came and brought eternal righteousness to those who take part in the covenant of his blood, though Messiah makes us clean, we still have to continue in the paths of righteousness or else we become wicked and one of the iniquity working fools.
When Abraham asked YHWH if he would kill the righteous with the wicked at Sodom and Gomorrah at Genesis 18, YHWH did not say "Nobody is righteous Abraham" But Abraham said "Far be it from You to do such a thing, to slay the righteous with the wicked so that the righteous and the wicked are treated alike"(Genesis 18:25);
I will list some scriptures showing righteous men, will add these soon.