Not Quite. But it is large number.
First there is the question of how to interpret "Household firearms". While Technically this leaves open the possibility that the quote is referring to all personal firearms; If we go with this definition we will include pretty much every single death by firearm outside of military and police operations. But personally, based on the words used I believe that they meant people killed by firearms owned and/or stored in their very own homes (which is a very common thing to talk about for the anti-gun crowd).
There are just over 30,000 deaths by firearms per year in the US, or just under 3K per five weeks. So as per the specific claim, I am sure there exist five week stretches that involved 3,400 deaths.
But not all firearm deaths are caused by firearms from the victims own house. The majority of deaths are suicides - around 16-21,000 per year. (I don't have any data on this, but I am taking it as a given that the vast majority of firearm assisted suicides are performed with a firearm they or a member of their family owns.)
Accidents are insignificant, making up just 1.5% of firearm related deaths.
Homicides make up around a third of all firearm related deaths. Some number of these will also be committed with household firearms, but I have been unable to find any specific statistics for this.
Source: Gun Policy.org