Change character limit of SQL Agent Operators
Just because you can does not mean you should. Please do not apply the changes below. This answer is only to show you why you should not change system tables or system procedures and what can go wrong.

msdb.dbo.sp_add_operator

When adding the operator with the ssms gui, msdb.dbo.sp_add_operator is called behind the scenes.

The email address parameter used is also nvarchar(100).



In theory, you could adapt the procedure by altering it

And then changing the @email_address parameter to nvarchar(1000)



If I then run it wih 384 characters (non gui)

USE [msdb]
GO
EXEC msdb.dbo.sp_add_operator @name=N'test', 
        @enabled=1, 
        @pager_days=0, 
        @email_address=N'emaione@example.com;emailtwo@example.com;emailthree@example.com;emaione@example.com;emailtwo@example.com;emailthree@example.com;emaione@example.com;emailtwo@example.com;emailthree@example.com;emaione@example.com;emailtwo@example.com;emailthree@example.com;emaione@example.com;emailtwo@example.com;emailthree@example.com;emaione@example.com;emailtwo@example.com;emailthree@example.com;'
GO


Getting the length:

select LEN(email_address)  as email_length
from sysoperators 
where name = 'test';


Shows the 384 characters are there.

email_length
384


However, one of the things instantly breaking is the operator list when opening it via ssms:



Due to ssms running this query:

create table #tmp_sp_help_operator
(id int null, name nvarchar(128) null, enabled tinyint null, email_address nvarchar(100) null, last_email_date int null, last_email_time int null, pager_address nvarchar(100) null, last_pager_date int null, last_pager_time int null, weekday_pager_start_time int null, weekday_pager_end_time int null, saturday_pager_start_time int null, saturday_pager_end_time int null, sunday_pager_start_time int null, sunday_pager_end_time int null, pager_days tinyint null, netsend_address nvarchar(100) null, last_netsend_date int null, last_netsend_time int null, category_name nvarchar(128) null)
insert into #tmp_sp_help_operator exec msdb.dbo.sp_help_operator



SELECT
tsho.name AS [Name],
'Server[@Name=' + quotename(CAST(
        serverproperty(N'Servername')
       AS sysname),'''') + ']' + '/JobServer' + '/Operator[@Name=' + quotename(tsho.name,'''') + ']' AS [Urn],
CAST(tsho.enabled AS bit) AS [Enabled]
FROM
#tmp_sp_help_operator AS tsho
ORDER BY
[Name] ASC

drop table #tmp_sp_help_operator


Showing why we should not play around with system tables &amp; procedures.

Who knows what will work and what will break when changing the table &amp; the procedure. 

A distribution list as vonPryz mentioned would be a much better solution.