I have a gutter leaking. The start price is $450 if I hire a company. I wonder if I could do it by myself but not sure what materials.
I saw the leaking is from the joint section of the gutter and the wall.
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Looks like a doug fir wood gutter, which would be consistent with the vintage brick, vintage wood shutters, and non-planar roof. Also, there is no end cap visible which is how vinyl and aluminum gutters are terminated.
The proper way to terminate the end of a wood gutter is with lead flashing contoured and sloped, let in with a chisel so that it is flush. After chiseling out, asphalt roofing (comes in cans or tubes) is applied to the newly exposed wood in order to bed the lead, and secured with copper tacks, then the seem and tack heads are wiped with a thin layer of asphalt. For all this, you'll might want to wear some gloves, but since you are not doing this regularly, I doubt gloves matter health wise.
Your gutter may have been installed this way, in which case you just need to de-tack, lift up the lead, remove/scrap and old asphalt that you can, apply new asphalt, and re tack it. This is common maintenance for wood gutters. It's not hard, but if you are not up to it, your only choice is to get someone else to do it, providing they know what required.
However, it looks like the gutter is attached directly to the fascia. That's a no-no for wood gutters and not a good idea for any type of gutter. The proper way is to space it off 1/4" or so, to give the back side of the gutter breathing room. That fact that it isn't suggests that the end was not terminated properly in the first place, so you'll be working from scratch. Another indication that it is not terminated properly is the absence of a wrap-over of the lead along the top lip.
So if it is attached directly to fascia, you'll want to schedule and budget for the gutters to be removed and reinstalled properly within the next couple of years. This will likely require a new course of drip edge backed by a 1x2 or similar nailed directly to the fascia. Otherwise, the rafter tails etc will rot along with the fascia.