The size rules on this line
Bags up to 160 cm in total (height plus width plus depth) travel free and fit in the overhead racks or at your feet.
Bags from 161 to 250 cm need a seat with an oversized baggage area. Bags over 250 cm are not allowed on board.
The rule applies to Nozomi, Hikari and Kodama trains alike, so there is no workaround by picking a slower train.
With a Japan Rail Pass, seats with an oversized baggage area are included at no extra cost. Book them by scanning the QR code on your pass at a Shinkansen ticket machine or at any staffed JR desk. Get your Japan Rail Pass
Which seats to book
The oversized baggage area is the space behind the last row of seats in each car. Book a seat in that last row and the space behind it is yours.
Some trains also offer reservable baggage compartments near the doors. If they are offered when you book, they work just as well.
Timing and crowds
Morning trains out of Tokyo and Friday evening trains are the busiest. Oversized baggage seats sell out first, so book as soon as you know your train.
During Golden Week, Obon and New Year, book days in advance. Non-reserved cars are not an option if your bag is over 160 cm.
Arriving in Kyoto or Osaka
Kyoto Station and Shin-Osaka have coin lockers, but large lockers fill up by late morning. Consider sending your bags ahead with a luggage forwarding service if you are doing a day trip.