ASHKELON, ISRAEL -- Sirens wail, radio broadcasts are interrupted, cellphones beep with Red Alerts every few seconds, and warning messages flash up on TV. When you hear them, rush for cover.

This has become the routine across large areas of central and southern Israel, from small towns bordering Gaza to metropolitan Tel Aviv and southern Beersheva.

More than 2,000 rockets have been launched by Hamas and other armed Palestinian groups into Israel this week alone, amid the most serious fighting between Israel and Gaza militants since 2014.

It's not a routine you can ever get used to, said Lior Dabush from the coastal city of Ashkelon, about 12 kilometres north of Gaza.

"We rarely leave the house," Dabush, 37, said from her apartment's 'safe' room - a mandatory feature for all new homes in Israel - where she now sleeps with her two children.

"We take short showers and we don't venture far from home," she told Reuters. "At times my eight-year-old son does not want to leave the safe room."

Hamas, the Islamist group that rules Gaza, launched the latest round of rocket attacks on Monday, after widespread Palestinian anger at threatened evictions of families from East Jerusalem, and Israeli police clashes with worshippers near Al-Aqsa Mosque, Islam's third holiest site.

The threat of rockets even penetrated as far as Jerusalem, when impacts in villages on the outskirts set off sirens in the center of the city, forcing Israelis taking part in an annual holiday to flee for cover, some running under the medieval battlements of the Jaffa Gate entrance to the walled Old City.

'DAMAGED FOR LIFE'

On the other side of the Gaza border, Palestinian civilians also find themselves trapped between the militant groups firing the rockets and the Israeli military, which has spent days bombarding Gaza with hundreds of aerial and artillery shells.

Residents of north Gaza have fled their homes to take shelter in UN-run schools and Palestinian officials say at least 124 people there have been killed, including 31 children.

The rockets have killed eight people in Israel, including a five-year-old who was struck by shrapnel that managed to pierce the shielding of his reinforced 'safe' room.

That attack happened in the border town of Sderot, where it's a matter of seconds between siren and impact, and the streets are largely empty of pedestrians.

Idit Botera, mother of a one-year-old child, said her sixth-floor apartment was damaged in the same barrage on Wednesday.

"We still haven't processed what happened, our blood is boiling," she said shortly after the strike. "These are kids who are being damaged for life and it doesn't make sense."

The impact on children - and the effect it will have on them later in life - is a common theme for Israelis living near Gaza, for whom rockets are an unwelcome but unavoidable fact of life.

In Netiv Haasara, a small Israeli community just north of the barrier that separates Israel from the Gaza Strip, tour guide Raz Shmilovitch, 45, reflected earlier this week on the toll the latest hostilities were taking.

"My family is not here now, I have taken them to a more remote safe place in which they will be safer to stay," he said.

"In the longer run, once the war is over we are going to have to deal with the consequences of raising post-traumatic kids," he told Reuters.

"If you have been living all your life as a kid under the threat of rockets being launched and being landed in your back yard, and you have between five to seven seconds from alarm to impact and that is the reality you are used to, that messes with your brain."

Reporting by Ran Tzabari, Amir Cohen, Baz Ratner and Ari Rabinovitch Editing by Gareth Jones.